Philippe Jouan

Immersion of Nonlinear Systems into Linear Systems Modulo Output Injection

The problem of the immersion of a SISO system into a linear up to an output injection one is studied in order to design Luenberger-like observers. Necessary and sufficient conditions are stated within a very general framework. Effective computations and examples are then provided.

Finite singularities of nonlinear systems. Output stabilization, observability and observers

In this paper, for the purposes, first, of constructing nonlinear observers and, second, of output stabilization for observed nonlinear systems, we develop a theory allowing one to deal with singularities that can appear. In the uncontrolled analytic case, we are especially interested in finite singularities.Using this theory, we generalize some of our previous results on the construction of nonlinear observers.Next, we consider the theorem of a very interesting paper [26] and similar theorems of one of the authors. We prove a result containing all these previous ones; this result allows us to stabilize, via dynamic output feedback, certain nonlinear systems that were only state feedback stabilizable.

Geometry of the Limit Sets of Linear Switched Systems

This paper is concerned with asymptotic stability properties of linear switched systems. Under the hypothesis that all the subsystems share a nonstrict quadratic Lyapunov function, we provide a large class of switching signals for which a large class of switched systems are asymptotically stable. For this purpose we define what we call nonchaotic inputs, which generalize the different notions of inputs with dwell time. Next we turn our attention to the behavior for possibly chaotic inputs. Finally, we give a sufficient condition for a system composed of a pair of Hurwitz matrices to be asymptotically stable for all inputs.

Immersion of nonlinear systems into linear systems modulo output injection

This paper deals with the problem of the immersion of a SISO system into a linear up to an output injection one. For this class of systems Luenberger-like observers can be designed: the dynamics of the error between the states of the observer and the immersed controlled system is linear. Necessary and sufficient conditions are first stated within a very-general framework. Effective computations and examples are then provided for uncontrolled and control-affine systems. In particular the control-affine case is completely (with a very slight restriction) solved.

Almost-Riemannian Geometry on Lie Groups

A simple Almost-Riemmanian Structure on a Lie group G is defined by a linear vector field and dim(G)-1 left-invariant ones. We state results about the singular locus, the abnormal extremals and the desingularization of such ARSs, and these results are illustrated by examples on the 2D affine and the Heisenberg groups.These ARSs are extended in two ways to homogeneous spaces, and a necessary and sufficient condition for an ARS on a manifold to be equivalent to a general ARS on a homogeneous space is stated.

EMPIRICAL INVARIANCE PRINCIPLE FOR ERGODIC TORUS AUTOMORPHISMS: GENERICITY

We consider the dynamical system given by an algebraic ergodic automorphism T on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process (f◦Ti)i∈ℕ, where f is a given ℝ-valued function. We give a sufficient condition on f for this Central Limit Theorem to hold. In the second part, we prove that the distribution function of a Morse function is continuously differentiable if the dimension of the manifold is at least three and Holder continuous if the dimension is one or two. As a consequence, the Morse functions satisfy the empirical invariance principle, which is therefore generically verified.

C^\infty and L^\infty Observability of Single-Input C^\infty -Systems

For single-input multi-output C∞-systems, we state conditions under which observability for every C∞-input implies observability for every almost everywhere continuous, bounded input (for every L∞-input in the control-affine case). A normal system is then defined as a system whose only bad inputs are smooth on some nonempty open set. When the state space is compact, normality turns out to be generic and enables us to extend some results of genericity of observability to nonsmooth inputs.

Singular linear systems on Lie groups; equivalence

Abstract In this paper, we define different kinds of singular systems on Lie groups and we analyze some of them. Furthermore, we state sufficient conditions for a general nonlinear singular system defined on a manifold to be equivalent by diffeomorphism to one of these models. Some examples are computed.

Convergence Rate of Nonlinear Switched Systems

This paper is concerned with the convergence rate of the solutions of nonlinear switched systems. We first consider a switched system which is asymptotically stable for a class of inputs but not for all inputs. We show that solutions corresponding to that class of inputs converge arbitrarily slowly to the origin. Then we consider analytic switched systems for which a common weak quadratic Lyapunov function exists. Under two different sets of assumptions we provide explicit exponential convergence rates for inputs with a fixed dwell-time.

LIS systems: The MIMO case

Abstract A system is LIS if it can be immersed into a linear up to an output injection one. For this class of systems Luenberger-like observers can be designed: the dynamics of the error between the states of the observer and the immersed controlled system is linear. The SISO case was previously studied and the present paper deals with the MIMO one. Necessary and sufficient conditions are first stated within a very general framework. Some computations are then provided for control-affine systems.